# Kcross

0th

Percentile

##### Multitype K Function (Cross-type)

For a multitype point pattern, estimate the multitype $K$ function which counts the expected number of points of type $j$ within a given distance of a point of type $i$.

Keywords
spatial, nonparametric
##### Usage
Kcross(X, i, j, r=NULL, breaks=NULL, correction,
..., ratio=FALSE, from, to )
##### Arguments
X
The observed point pattern, from which an estimate of the cross type $K$ function $K_{ij}(r)$ will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.
i
The type (mark value) of the points in X from which distances are measured. A character string (or something that will be converted to a character string). Defaults to the first level of marks(X).
j
The type (mark value) of the points in X to which distances are measured. A character string (or something that will be converted to a character string). Defaults to the second level of marks(X).
r
numeric vector. The values of the argument $r$ at which the distribution function $K_{ij}(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important
breaks
This argument is for internal use only.
correction
A character vector containing any selection of the options "border", "bord.modif", "isotropic", "Ripley", "translate", "translation", "none" or
...
Ignored.
ratio
Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.
from,to
An alternative way to specify i and j respectively.
##### Details

This function Kcross and its companions Kdot and Kmulti are generalisations of the function Kest to multitype point patterns.

A multitype point pattern is a spatial pattern of points classified into a finite number of possible colours'' or types''. In the spatstat package, a multitype pattern is represented as a single point pattern object in which the points carry marks, and the mark value attached to each point determines the type of that point. The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to as.ppp. It must be a marked point pattern, and the mark vector X$marks must be a factor. The arguments i and j will be interpreted as levels of the factor X$marks. If i and j are missing, they default to the first and second level of the marks factor, respectively. The cross-type'' (type $i$ to type $j$) $K$ function of a stationary multitype point process $X$ is defined so that $\lambda_j K_{ij}(r)$ equals the expected number of additional random points of type $j$ within a distance $r$ of a typical point of type $i$ in the process $X$. Here $\lambda_j$ is the intensity of the type $j$ points, i.e. the expected number of points of type $j$ per unit area. The function $K_{ij}$ is determined by the second order moment properties of $X$.

An estimate of $K_{ij}(r)$ is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the process of type $i$ points were independent of the process of type $j$ points, then $K_{ij}(r)$ would equal $\pi r^2$. Deviations between the empirical $K_{ij}$ curve and the theoretical curve $\pi r^2$ may suggest dependence between the points of types $i$ and $j$.

##### References

Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.

Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.

Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants' nests. Applied Statistics 32, 293--303 Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several types of points. J. Royal Statist. Soc. Ser. B 44, 406--413.

Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.

Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition. Springer Verlag, 1995.

Kdot, Kest, Kmulti, pcf

• Kcross
##### Examples
# amacrine cells data
K01 <- Kcross(amacrine, "off", "on")
plot(K01)

<testonly>K01 <- Kcross(amacrine, "off", "on", ratio=TRUE)</testonly>
K10 <- Kcross(amacrine, "on", "off")

# synthetic example: point pattern with marks 0 and 1
pp <- runifpoispp(50)
pp <- pp %mark% factor(sample(0:1, npoints(pp), replace=TRUE))
K <- Kcross(pp, "0", "1")
K <- Kcross(pp, 0, 1) # equivalent
Documentation reproduced from package spatstat, version 1.42-2, License: GPL (>= 2)

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